Another drastic example of animal sleep being very different from ours is dolphins. Dolphins sleep "one hemisphere at a time".

> Research has shown that dolphins are able to sleep with only half of their brain at a time, a phenomenon known as unihemispheric slow-wave sleep (USWS). During USWS, one hemisphere of the brain remains active while the other hemisphere rests. This allows dolphins to continue swimming, surfacing for air, and avoiding predators while still getting the rest they need.

Try that! :-)